In this appendix we discuss these two scaling regimes,
the paraxial white-noise and scintillation regimes, and the relation to the Itô-Schrödinger equation, and we refer to [1]}, [2]} for the full derivation.
Consider \(\hat{u}(z,{{\itbf x}})\)  satisfying the Helmholtz equation
(REF ).
Let \(\sigma \)  be the standard deviation of the fluctuations of the index of refraction
\(n\)  in this equation. Moreover, assume here that the random fluctuations of the index of
refraction is isotropic and denote by \(l_{\rm c}\)  the correlation length of the fluctuations,
by \(\lambda \)  the wavelength,
by \(L\)  the typical propagation distance,
and by \(r_o\)  the transverse radius of the initial beam,
which in this paper corresponds to the dimension of the time-reversal mirror.
We introduce the wavenumber defined by
\(k = \frac{\omega }{c_o} = \frac{2\pi }{\lambda },\) 
